set-alias installutil $env:windir\Microsoft.NET\Framework\v2.0.50727\installutil.exe
installutil 'C:\Progra~1\Veeam\Backup and Replication\Backup\Veeam.Backup.PowerShell.dll'
Microsoft (R) .NET Framework Installation utility Version 2.0.50727.8007
Copyright (c) Microsoft Corporation. All rights reserved.
Exception occurred while initializing the installation:
System.BadImageFormatException: Could not load file or assembly 'file:///C:\Program Files\Veeam\Backup and Replication\B
ackup\Veeam.Backup.PowerShell.dll' or one of its dependencies. This assembly is built by a runtime newer than the curren
tly loaded runtime and cannot be loaded..

As far as I know, uselegacyv2runtimeactivationpolicy should be enabled and both 2.0.50727 and 4.0.30319 versions should be listed as supported runtimes. Those changes should be reflected in PS configuration file. But that's more like a first guess. Thanks.

Windows PowerShell
Copyright (C) 2014 Microsoft Corporation. All rights reserved.
PS C:\Windows\system32> C:\Windows\Microsoft.NET\Framework64\v4.0.30319\installutil.exe "c:\progra~1\veeam\backup~1\back
up\Veeam.Backup.PowerShell.dll"
Microsoft (R) .NET Framework Installation utility Version 4.0.30319.33440
Copyright (C) Microsoft Corporation. All rights reserved.
Running a transacted installation.
Beginning the Install phase of the installation.
See the contents of the log file for the c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.dl
l assembly's progress.
The file is located at c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.InstallLog.
Installing assembly 'c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.dll'.
Affected parameters are:
logtoconsole =
assemblypath = c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.dll
logfile = c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.InstallLog
An exception occurred while trying to find the installers in the c:\Program Files\Veeam\Backup and Replication\backup\Ve
eam.Backup.PowerShell.dll assembly.
System.Reflection.ReflectionTypeLoadException: Unable to load one or more of the requested types. Retrieve the LoaderExc
eptions property for more information.
Aborting installation for c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.dll.
An exception occurred during the Install phase.
System.InvalidOperationException: Unable to get installer types in the c:\Program Files\Veeam\Backup and Replication\bac
kup\Veeam.Backup.PowerShell.dll assembly.
The inner exception System.Reflection.ReflectionTypeLoadException was thrown with the following error message: Unable to
load one or more of the requested types. Retrieve the LoaderExceptions property for more information..
The Rollback phase of the installation is beginning.
See the contents of the log file for the c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.dl
l assembly's progress.
The file is located at c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.InstallLog.
Rolling back assembly 'c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.dll'.
Affected parameters are:
logtoconsole =
assemblypath = c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.dll
logfile = c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.InstallLog
An exception occurred while trying to find the installers in the c:\Program Files\Veeam\Backup and Replication\backup\Ve
eam.Backup.PowerShell.dll assembly.
System.Reflection.ReflectionTypeLoadException: Unable to load one or more of the requested types. Retrieve the LoaderExc
eptions property for more information.
Aborting installation for c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.dll.
An exception occurred during the Rollback phase of the System.Configuration.Install.AssemblyInstaller installer.
System.InvalidOperationException: Unable to get installer types in the c:\Program Files\Veeam\Backup and Replication\bac
kup\Veeam.Backup.PowerShell.dll assembly.
The inner exception System.Reflection.ReflectionTypeLoadException was thrown with the following error message: Unable to
load one or more of the requested types. Retrieve the LoaderExceptions property for more information..
An exception occurred during the Rollback phase of the installation. This exception will be ignored and the rollback wil
l continue. However, the machine might not fully revert to its initial state after the rollback is complete.
The Rollback phase completed successfully.
The transacted install has completed.
The installation failed, and the rollback has been performed.
PS C:\Windows\system32>

Affected parameters are:
logtoconsole =
assemblypath = c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.dll
logfile = c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.InstallLog
An exception occurred while trying to find the installers in the c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.dll assembly.
System.Reflection.ReflectionTypeLoadException: Unable to load one or more of the requested types. Retrieve the LoaderExceptions property for more information.
Aborting installation for c:\Program Files\Veeam\Backup and Replication\backup\Veeam.Backup.PowerShell.dll.

Hi all,
i looked at C:\Program Files\Veeam\Backup and Replication\Backup and there are no Veeam.Backup.PowerShell.dll. version Veeam Free 8.0.0.2030
There are only Veeam.Backup.SanPlugin.Hp3PAR.PowerShell.dll, Veeam.Backup.SanPlugin.HpP4k.PowerShell.dll, Veeam.Backup.SanPlugin.NetApp.PowerShell.dll, Veeam.Backup.PSService.dll
which of dll i have to register?

Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\PowerShell\1\PowerShellSnapIns\VeeamPSSnapIn]
"PowerShellVersion"="2.0"
"Vendor"="Veeam"
"Description"="This is a PowerShell snap-in that includes the Veeam's cmdlet."
"VendorIndirect"="VeeamPSSnapIn, Veeam"
"DescriptionIndirect"="VeeamPSSnapIn,This is a PowerShell snap-in that includes the Veeam's cmdlet."
"Version"="8.0.0.0"
"ApplicationBase"="C:\\Program Files\\Veeam\\Backup and Replication\\Backup"
"AssemblyName"="Veeam.Backup.PowerShell, Version=8.0.0.0, Culture=neutral, PublicKeyToken=bfd684de2276783a"
"ModuleName"="C:\\Program Files\\Veeam\\Backup and Replication\\Backup\\Veeam.Backup.PowerShell.dll"

Amazing how much BS there is about getting this to work and something like this is what it takes. Run the installer from the original media and then create a .REG file with this code and merge it with your registry. Thanks so much for posting this.